Does $\operatorname{Hom}_A (A / P, M) \not = 0$ imply that $P$ is an associated prime of $M$?

Question

$A$ is Noetherian, $M$ is finitely generated.

Does $\operatorname{Hom}_A (A / P, M) \not = 0$ imply that $P$ is an associated prime of $M$?

I am trying to prove that associated primes respect localization, and I've reduced it to this step (following the hint in Ravi). It follows that there is some $m \in M$ so that $\operatorname{ann}(m) \supseteq P$, but I don't see how to get exact equality.

Answer 1

$\operatorname{Hom}_\mathbb{Z}(\mathbb{Z}/(0),\mathbb{Z}/(2))\neq 0$, but $(0)$ is not an associated prime of $\mathbb{Z}/(2)$.